The wave nature of electrons and the periodic lattice of atoms give rise to allowed energy bands and forbidden energy gaps for electrons in a solid. The forbidden gaps arise from the destructive interference of electrons for certain wavelengths and directions. If a forbidden gap exists for all possible directions, it is referred to as a complete bandgap. A semiconductor has a complete bandgap between the valence and conduction bands.
The optical analogy is the photonic crystal, where a periodic lattice of contrasting dielectric structures (i.e., different indices of refraction) provides the periodic potential for light that atoms do for electrons. Photonic crystals can be thought of as extensions of diffraction gratings (i.e., a one-dimensional photonic crystal) or naturally occurring crystals used in X-ray crystallography. Light interacting with a diffraction grating or X-ray crystal interacts with the periodic structure and is redistributed into “allowed” and precluded from “forbidden” directions. The forbidden directions are the “photonic bandgaps” of the structure.
Photonic crystals can be designed with photonic bandgaps that prevent light of a certain wavelength and direction from propagating within the photonic crystal. If the photonic crystal does not allow light to propagate within a wavelength range for all polarizations and directions, it is said to have a “complete photonic bandgap.” A necessary condition for a complete photonic bandgap is that the contrasting dielectric lattice be periodic in three dimensions (3D).
Research of photonic crystals and their behavior was prompted by the article by Yablonovitch, entitled “Inhibited spontaneous emission in solid-state physics and electronics,” in Phys. Rev. Lett. Vol. 58, No. 20, 2059–2062 (1987). Based on theoretical considerations, a number of new optical devices, from better lasers to extremely miniaturized light switches and guides, have been suggested by workers in this relatively new field.
While photonic crystals offer a great deal of promise in fabricating new devices, fabricating such crystals with predetermined structures is daunting. The article by Yablonovitch et al., entitled “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” in Phys. Rev. Lett. Vol.67, No. 17,2295–2298 (1991), describes the formation of the first artificial 3D photonic crystal by drilling an array of intersecting millimeter size holes in a dielectric material. This photonic crystal has a bandgap in the microwave range of the spectrum and is of limited practical interest.
Since the early pioneering work by Yablonovitch, a great deal of research has been devoted to the fabrication and study of photonic crystals in the infrared and visible. The article by Bimer et al., entitled “Silicon-based photonic crystals,” in Adv. Mater. Vol. 13, No. 6, Mar. 16, 2001, describes fabricating two-dimensional (2D) and 3D photonic crystals. 2D photonic crystals have periodicity in two dimensions and are uniform in the third dimension and are much easier to fabricate than 3D photonic crystals. Although a 2D photonic crystal can not have a complete bandgap in the strictest sense, it can have a forbidden gap that exists for all directions and polarizations of propagation precisely confined to the plane of periodicity. In this more limited sense, the forbidden gap is referred to as a “complete 2D bandgap.”
While 2D photonic crystal waveguides are useful for certain applications such as planar circuits and distributed feedback (DFB) lasers, there are a host of other applications (e.g., the formation of ultra-small optical and electro-optical integrated circuits and devices) that call for 3D photonic crystal waveguides. To date, however, readily forming 3D photonic crystals waveguides has proven difficult. This is particularly true where the desired bandgap wavelength is at the optical or infrared, since the dimensions of the lattice must be a fraction of the bandgap wavelength.
One application for a 3D photonic crystal having a complete bandgap is to guide light. This can be accomplished by carving a path into such a photonic crystal to serve as an air-filled waveguide. Light that propagates in the air-filled waveguide at a frequency within the complete bandgap will be totally reflected by the photonic crystal and be totally confined to and directed along the waveguide. It should confine light around tight bends much better than conventional waveguides (e.g., optical fibers), where the guiding depends on the limited angular range of total internal reflection at the interface between the higher index core and the lower index cladding.
While some techniques have been developed for fabricating 3D photonic crystals, they involve extreme process conditions, such as forming individual dielectric layers and then stacking and bonding the layers to build the crystal. The formation of 3D waveguides in such crystals adds yet another level of complexity.
Whether a particular 3D photonic crystal has a complete or incomplete photonic bandgap depends on the particular lattice type and its parameters (e.g., the size, shape and spacing of the voids). Fortunately, the bandgap properties of a photonic crystal formed from a given lattice type (e.g., diamond) can be calculated. One technique for determining bandgap properties of a 3D photonic crystal based on the lattice parameters is presented in the article by Ho et al., entitled “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett., Vol. 65, No. 25, 3152–3155 (1990), which article is incorporated by reference herein. The Ho article shows that certain crystals lattices (e.g., diamond) can form 3D photonic crystals having either a complete or incomplete bandgap, depending on the above-mentioned lattice parameters.
Given the increasing importance of 3D complete bandgap photonic crystals, it would be desirable to have a method of modifying the lattice of a incomplete bandgap 3D photonic crystal to form a complete bandgap 3D photonic crystal.